The complete dynamics description of positively curved metrics in the Wallach flag manifold $\mathrm{S}\mathrm{U}(3)\mathrm{/}{\mathrm{T}}^2$

The family of invariant Riemannian manifolds in the Wallach flag manifold $\mathrm{S}\mathrm{U}(3)\mathrm{/}{\mathrm{T}}^2$ is described by three parameters $(x,y,z)$ of positive real numbers. By restricting such a family of metrics in the \emph{tetrahedron} $\mathcal{T}:=\mathcal{x}+\mathcal{y}+\mathcal{z}=\mathcal{1}$, in this paper, we describe all regions $\mathcal{R}\subset \mathcal{T}$ admitting metrics with curvature properties varying from positive sectional curvature to positive scalar curvature, including positive intermediate curvature notion’s. We study the dynamics of such regions under the \emph{projected Ricci flow} in the plane $(x,y)$, concluding sign curvature maintenance and escaping. In addition, we obtain some results for positive intermediate Ricci curvature for a path of metrics on fiber bundles over $\mathrm{S}\mathrm{U}(3)\mathrm{/}{\mathrm{T}}^2$, further studying its evolution under the Ricci flow on the base.